Photocurrent-based detection of Terahertz radiation in grapheneAndrea Tomadin,1, a) Alessandro Tredicucci,1 Vittorio Pellegrini,2, 1 Miriam S. Vitiello,1 and Marco Polini11)NEST, Istituto Nanoscienze-CNR and Scuola Normale Superiore, I-56126 Pisa,Italy2)Istituto Italiano di Tecnologia (IIT), Via Morego 30, 16163 Genova, Italy

Graphene is a promising candidate for the development of detectors of Terahertz (THz) radiation. A well-known detection scheme due to Dyakonov and Shur exploits the confinement of plasma waves in a field-effecttransistor (FET), whereby a dc photovoltage is generated in response to a THz field. This scheme has alreadybeen experimentally studied in a graphene FET [L. Vicarelli et al., Nature Mat. 11, 865 (2012)]. In the questfor devices with a better signal-to-noise ratio, we theoretically investigate a plasma-wave photodetector inwhich a dc photocurrent is generated in a graphene FET. The rectified current features a peculiar change ofsign when the frequency of the incoming radiation matches an even multiple of the fundamental frequency ofplasma waves in the FET channel. The noise equivalent power per unit bandwidth of our device is shown tobe much smaller than that of a Dyakonov-Shur detector in a wide spectral range.

In a series of pioneering papers1–4, Dyakonov and Shurproposed a mechanism enabling detection of Terahertz(THz) radiation, which is based on the fact that a field-effect transistor (FET) hosting a two-dimensional (2D)electron gas acts as a cavity for plasma waves. In theDyakonov-Shur (DS) scheme plasma waves are launchedby modulating the potential difference between gate andsource. When a plasma wave launched at the sourcecan reach the drain in a time shorter than the momen-tum relaxation time, the detection of radiation exploitsconstructive interference in the cavity. In this case oneachieves frequency-resolved detection of the incoming ra-diation (“resonant regime”). Broadband detection occurswhen plasma waves are overdamped or when the lengthof the FET channel is larger than the length over whicha plasma wave can travel.

In the DS detection scheme a dc photovoltage is gener-ated between source and drain in response to the incom-ing oscillating field. In the resonant regime, the dc pho-toresponse is characterized by peaks at the odd multiplesof the lowest plasma-wave frequency. For typical devicelengths and carrier densities, the fundamental plasma-wave frequency νP is in the THz range, so that photode-tectors based on the DS mechanism are naturally usefulin the context of THz detection. We emphasize that asubstantial amount of experimental work has been car-ried out on DS photodetection in ordinary (III,V) semi-conductors5,6. For the sake of completeness, we pointout that resonant excitation and detection of 2D and 3Dplasma-wave oscillations has been demonstrated also out-side of the DS scheme. For example, in Ref. 7 interbandphotoexcitation is used to launch a plasma wave, insteadof modulating the potential difference between gate andsource, while in Ref. 8 the change in resistance betweensource and drain is measured, instead of a dc photovolt-

a)Electronic mail: andrea.tomadin@sns.it

age.Recently, it has been understood that graphene can

pave the way for the realization of robust and cheap THzdetectors operating at room temperature and based onthe DS scheme9,10. Graphene, a 2D crystal of Carbonatoms packed in a honeycomb lattice11,12, has indeedhigh carrier mobility, even at room temperature, a gap-less spectrum, and a frequency-independent absorption,making it an ideal platform for a variety of applications inphotonics, optoelectronics, and plasmonics13,14. Vicarelliet al.9 have demonstrated room-temperature THz detec-tors based on antenna-coupled graphene FETs, which ex-ploit the DS mechanism but display also contributions ofphoto-thermoelectric origin. The plasma waves excited

Ua(t)

I

radiation

U0L

Wg

d

FIG. 1. Schematic representation of the setup studied in thiswork. The drain and source contacts of a graphene FET areconnected to the feeds of an antenna (not shown), which col-lects impinging THz radiation. An oscillating potential differ-ence is generated between drain and source, which is kept atconstant potential U0 with respect to a back gate. The latteris confined in the region below the sample to avoid detrimen-tal effects on the antenna operation. A dc photocurrent I,which is proportional to the power of the incoming radiation,is generated between source and drain.

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by THz radiation in Ref. 9 are overdamped and the fabri-cated detectors, although enabling large area, fast imag-ing of macroscopic samples, do not yet operate in theaforementioned resonant regime.

In this Letter we discuss a graphene photodetector inwhich a dc photocurrent appears in response to an os-cillating field, which is fed between source and drain bythe lobes of an antenna. This coupling geometry betweenantenna lobes and FETs has already been experimentallyimplemented15 in AlGaAs/InGaAs/AlGaAs heterostruc-tures. Note that in Ref. 15 the oscillations of the gate-to-channel potential play a crucial role in the rectificationof the incoming signal. On the contrary, in our setup,the gate potential is constant (i.e. it merely fixes the av-erage carrier density) and screens the long-range tail ofthe carrier-carrier Coulomb interaction. This allows us touse the hydrodynamic theory to describe propagation ofplasma waves in the device. Moreover, the setup that wediscuss differs from that realized in Ref. 15 because therectification which produces the dc photocurrent stemsfrom the intrinsic nonlinear nature of the hydrodynamictheory. The origin of the nonlinear response is thus thesame as in the standard DS scheme. Finally, the op-erating principle of the device proposed in this Lettercontrasts the one proposed in Ref. 16, where source anddrain are also connected to the lobes of an antenna, butthe nonlinear element is a Schottky contact.

The graphene-based device proposed in this work canfunction as a broadband or resonant detector of THz ra-diation. In the latter case, the photocurrent as a functionof the frequency of the incoming radiation is character-ized by sharp peaks at even multiples of the fundamen-tal plasma frequency. Notably, the photocurrent changessign at these peaks.

Our analysis relies on the hydrodynamic theory17–21,i.e. on the combined use of continuity and Euler equa-tions. The continuity equation reads

∂tn(x, t) + ∂x[n(x, t)v(x, t)] = 0 , (1)

where n(x, t) is the electron density and v(x, t) is theelectron drift velocity. The x coordinate varies along thetransport direction in a field effect transistor geometry inwhich the source (drain) is placed at x = 0 (x = L)—seeFig. 1. We assume that the hydrodynamic variables donot depend on the direction perpendicular to transport.If the distance d between graphene and the gate is muchsmaller than the typical wavelength of plasma oscillationsin graphene, the following local relation (known as the“gradual channel approximation”) between the densityand the “gate-to-channel swing” U(x, t) holds21:

n(x, t) = C

eU(x, t) . (2)

Here, C = εsub/(4πd) is the geometrical capacitance perunit area, with εsub the dielectric constant of the insulatorseparating graphene from the gate, and e is the absolutevalue of the electron charge. The gate-to-channel swing

can be written in the form U(x, t) = U0 +δU(x, t), whereU0 is the gate-to-source potential difference.

We then employ the Euler equation of motion21

∂tv(x, t) + v(x, t)∂xv(x, t) = − e

mc∂xδU(x, t)

+ e

mc

12U0

δU(x, t)∂xδU(x, t)− 1τv(x, t) .

(3)

Here, mc = ~kF/vF is the cyclotron mass, vF ' 1 nm/fsis the Fermi velocity, and kF = (πn0)1/2 = (πCU0/e)1/2

is the Fermi wave number corresponding to the densityn0 = CU0/e fixed by the gate voltage U0. In writ-ing Eq. (3) we have neglected contributions21 due tothe pressure and corrections that are important whenthe Fermi velocity vF is comparable to the plasma wavespeed s = (eU0/mc)1/2. Note that Eq. (3) includes aphenomenological friction term, arising due to scatteringof electrons with impurites and phonons, which is pro-portional to the momentum relaxation rate τ−1.

We consider the setup illustrated in Fig. 1, where theantenna feeds are connected to source and drain. The in-coming radiation, with frequency Ω, generates a potentialdifference Ua(t) = εUa cos (Ωt) between the lobes of theantenna. We solve Eqs. (1) and (3) with the followingboundary conditions

U(0, t) = U0, U(L, t) = U0 + εUa cos (Ωt) , (4)

by utilizing a series expansion in the amplitude of the os-cillating perturbation. We stress that the dimensionlessparameter ε has been introduced to distinguish terms ofdifferent order in the series expansion. The final resultfor the dc photocurrent I, which is proportional to U2

a ,is obtained by letting ε→ 1. The sign convention for thephotocurrent is the following: I is positive when it flowsfrom source to drain.

We start by expanding the hydrodynamic variables ina power series:

v(x, t) = εv1(x, t) + ε2[δv(x) + v2(x, t)] , (5)U(x, t) = U0 + εU1(x, t) + ε2[δU(x) + U2(x, t)] . (6)

Here, Un(x, t) and vn(x, t) are periodic functions of timet with frequency nΩ, n = 0, 1, 2, . . . . To first order inε, the boundary conditions (4) become:

U1(0, t) = 0, U1(L, t) = εUa cos(Ωt) . (7)

The first-order solutions of Eqs. (1) and (3) can be easilydetermined and read as following:

v1(x, t) = Ua

2U0

ΩK

eiKx + e−iKx

eiKL − e−iKLe−iΩt + c.c. , (8)

U1(x, t) = Ua

2eiKx − e−iKx

eiKL − e−iKLe−iΩt + c.c. , (9)

with K = (Ω/s)√

1 + i/(Ωτ).

3

0 1 2 3 4 5

Ω/ωP

−3

−101

3

5I/I d

τ

FIG. 2. The photocurrent I in units of the diffusive cur-rent Id [defined in Eq. (16)], which is proportional to τ , isplotted as a function of the ratio between the frequency Ω ofthe incoming radiation and the fundamental plasma angularfrequency ωP = πs/(2L). Different curves correspond to dif-ferent values of the momentum relaxation time τ . The valuesof τ are 0.5 L/s, 1.0 L/s, 2.0 L/s, and 10.0 L/s. The arrowindicates increasing values of τ .

To second order in ε and after averaging over the periodT = 2π/Ω of the incoming radiation, we find:

∂x [U0δv(x) + 〈U1(x, t)v1(x, t)〉t] = 0 , (10)

〈v1(x, t)∂xv1(x, t)〉t = − e

mc∂xδU(x)− 1

τδv(x)

+ e

mc

12U0〈U1(x, t)∂xU1(x, t)〉t ,

(11)

with boundary conditions δU(0) = δU(L) = 0. In theprevious equations 〈f(t)〉t ≡ T−1 ∫ T

0 dtf(t) denotes aver-aging over time.

We now need to evaluate the current density J(x, t) =−en(x, t)v(x, t). The first non-zero contribution to thedc current density is of order ε2 and reads

J = 〈J(x, t)〉t = −ε2C [U0δv(x) + 〈U1(x, t)v1(x, t)〉t] .(12)

Comparing Eq. (12) with Eq. (10), we immediately con-clude that J is uniform in space, as expected. We nowsolve Eq. (12) for δv(x) and substitute the result inEq. (11). Finally, we integrate the resulting equationin space from x = 0 to x = L, using that J is uniform.The final result is:

J = −ε2CL

∫ L

0dx〈U1(x, t)v1(x, t)〉t

+ ε2CτU0

L

∫ L

0dx〈v1(x, t)∂xv1(x, t)〉t

− ε2 Cτe

2mcL

∫ L

0dx〈U1(x, t)∂xU1(x, t)〉t .

(13)

0 2 4 6 8 10

τs/L

−3

−101

3

5

I/I d

Ω

FIG. 3. The photocurrent I in units of the diffusive currentId is plotted as a function of the ratio between the momentumrelaxation time τ and L/s. Different curves correspond todifferent values of the incoming radiation frequency Ω. Thevalues of Ω range from 1.5 ωP to 2.0 ωP in steps of 0.025 ωP,increasing in the direction indicated by the arrow.

The time- and space-integrals in Eq. (13) can be readilyevaluated by employing Eqs. (8) and (9). The dc pho-tocurrent is given by I = Wg × J |ε=1, where Wg is thewidth of the device:

I = σ0U0

2Wg

L

(Ua

2U0

)2[1 + 2β(Ωτ)F (Ω, τ)] , (14)

where β(x) ≡ 2x/√

1 + x2 and

F (Ω, τ) = cosh (2K2L) + cos (2K1L)− 2cosh (2K2L)− cos (2K1L) . (15)

Here, K1 and K2 are the real and the imaginary part ofthe wave number K, which depend on Ω and τ . In thefinal expression we have introduced the Drude formulaσ0 = σ0(U0) = n0e

2τ/mc for the conductivity of 2DMDFs. Eq. (14) is the main result of this Letter. Wenote that in the limit Ω τ−1, which corresponds tothe limit of diffusive transport, Eq. (14) yields I → Idwith

Id ≡ σ0U0

2Wg

L

(Ua

2U0

)2. (16)

As expected, the photocurrent Id in the limit of diffusivetransport is proportional to the conductivity σ0, whichgrows linearly with τ . Note that Eq. (16) can be writtenas Id = Wgσ0∆Ud/L where ∆Ud = (Ua/2)2σ−1

0 dσ0/dU0coincides with the well-known DS photovoltage that ap-pears when the antenna feeds are connected to gate andsource9,22. The diffusive result (16) can also be obtainedby solving the continuity equation coupled to Ohm’slaw J(x, t) = σ0(U(x, t))∂xU(x, t) where σ0(U(x, t)) =σ0(U0)|U0→U(x,t).

Illustrative plots of the photocurrent I (in units ofId ∝ τ) are shown in Figs. 2 and 3. In the limit of

4

0.0 0.1 0.2 0.3

Ω/ωP

−3.0

−2.5

−2.0

−1.5

−1.0

−0.5lo

g10[N

EPI/N

EPV

]

τ

0.0 0.5 1.0 1.5 2.0

−2

0

2

FIG. 4. The ratio NEPI/NEPV (in logarithmic scale) asfrom Eq. (17) is plotted as a function of Ω (in units of ωP).Different curves correspond to different values of the momen-tum relaxation time τ . The values of τ are 10.0 L/s, 20.0 L/s,30.0 L/s. The arrow indicates increasing values of τ . The in-set shows the ratio NEPI/NEPV in a larger range of Ω andfor τ = 30.0 L/s. (The axis labels of the inset are not shownsince they are the same as in the main panel.) All the resultsshown in this Figure are restricted to the frequency domainwhere I < 0.

diffusive transport, the current is positive and frequencyindependent and its magnitude grows linearly with τ . Inthis regime, the device realizes a broadband photodetec-tor. It is important to notice that a finite dc current ispossible because the reflection symmetry x → −x is ex-plicitly broken by the boundary conditions (4). When τincreases, the current increases (decreases) in the neigh-borhood of the even (odd) multiples of the fundamentalplasma angular frequency ωP = πs/(2L). For typical de-vice lengths, the plasma-wave frequency νP = ωP/(2π)is in the THz regime23. The sign of the current becomesnegative in the windows of frequency between even mul-tiples of ωP. Eventually, for large τ , the current is con-stant and negative everywhere (I = −3 Id) except at evenmultiples of ωP, where sharp peaks with positive currentI = 5 Id appear. In this regime the propagation of plasmawaves is ballistic and the device realizes a resonant pho-todetector. Tuning of the gate voltage allows to changeωP and thus to measure Ω by detecting the sharp switchof the current direction. From Fig. 3 we see that thesharp peaks at Ω = 2nωP are a robust feature, whichpersists even for Ωτ 1.

We now proceed to compare the performance of ourdevice to that of a standard DS photodetector3 in whicha dc photovoltage is measured in response to the imping-ing radiation. Since the current and voltage responsivi-ties of the two setups have different physical dimensions,we compare the noise equivalent power (NEP) per unitbandwidth. The NEP per unit bandwidth has dimen-sions of W/Hz1/2 in both cases.

Let us assume that a power Pabs is collected by the

antenna. The relation between the potential differenceUa at the antenna feeds and Pabs can be parameterizedby |Ua|2 = gPabs, where g is the so-called coupling ef-ficiency. The current (voltage) responsivity is definedby RI ≡ |I|/Pabs (RV ≡ |∆U |/Pabs, where ∆U is theDS photovoltage3). The current noise In(∆f) and thevoltage noise ∆Un(∆f) in a bandwidth ∆f are relatedby In(∆f) = ∆Un(∆f)/R, where R = L/(Wgσ0) is thechannel resistance. The NEP in the current (voltage)setup NEPI [NEPV ] per unit bandwidth is defined by theratio In(∆f)/RI [Vn(∆f)/RV ]. We therefore find thatthe NEP per unit bandwidth of our device, measured inunits of the NEP per unit bandwidth of a DS photovolt-age detector in the usual configuration3, is given by thefollowing ratio:

NEPINEPV

= 2f(Ω)1 + 2β(Ωτ)F (Ω, τ) , (17)

where3

f(Ω) = 1 + β(Ωτ)− 1 + β(Ωτ) cos(2K1L)sinh2 (K2L) + cos2 (K1L)

, (18)

andK1, K2, β(Ωτ), and F (Ω, τ) have been defined above.We emphasize that NEPI/NEPV is independent of thecoupling efficiency g.

The ratio in Eq. (17) is plotted in Fig. 4 as a func-tion of Ω/ωP and for different values of τ . The NEPin our device, i.e. NEPI , is several orders of magnitudesmaller than NEPV for frequencies of the incoming ra-diation Ω ωP (main panel in Fig. 4). In this regime,the photocurrent is finite and equal to −3Id (in the bal-listic regime), while the DS photovoltage vanishes in thesame limit (see discussion in Sect. V B of Ref. 3). A sim-ilar advantage over the DS detection scheme has beennoted in Ref. 24, where the authors propose a photovolt-age detection scheme in which the antenna signal is fedto drain and source, as in our setup. From the inset inFig. 4 we see that the ratio of the two NEPs varies overseveral orders of magnitude and that NEPI NEPV forΩ ∼ 2nωP. The device proposed in this Letter therebyfeatures substantial advantages with respect to the stan-dard DS scheme in terms of signal-to-noise ratio in awide spectral range. Before concluding, we would like topoint out that for typical parameters23 and sufficientlyshort devices (L . 1 µm), the frequency range plotted inthe horizontal axis of the main panel of Fig. 4 is still inthe hundreds of GHz/THz range.

In conclusion, we have shown that a dc photocurrent isgenerated in a field-effect transistor when the incomingradiation is collimated on the device by connecting theantenna feeds to source and drain. The generated pho-tocurrent features a peculiar change of sign when the fre-quency of the radiation matches an even multiple of thefundamental frequency of plasma waves in the channel.We have carried out a detailed analysis for a graphenefield-effect transistor, finding a room-temperature noiseequivalent power which, at least in principle, can be com-parable to commercially available Terahertz detectors.

5

ACKNOWLEDGMENTS

We thank Michele Ortolani for useful discussions. Thiswork was supported by the EC under Graphene Flagship(contract no. CNECT-ICT-604391), the Italian Ministryof Education, University, and Research (MIUR) throughthe program “FIRB - Futuro in Ricerca 2010” GrantNo. RBFR10M5BT (“PLASMOGRAPH”) and GrantNo. RBFR10LULP (“FRONTERA”), and the ItalianMinistry of Economic Development through the ICE-CRUI project “TERAGRAPH”. We have made use offree software (www.gnu.org, www.python.org).

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23With the choice of parameters d ∼ 100 nm, kF ∼ 0.17 nm−1

(corresponding to a typical doping n0 ∼ 1012 cm−2) we finds ∼ 8.2 nm/fs. In this case, changing L from 300 nm to 3 µm,the plasma-wave frequency in a graphene FET changes from νP ∼6.8 THz to νP ∼ 680 GHz.

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